Option pricing: Basics

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Option pricing Basics

• Aswath Damodaran

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The ingredients that make an option

• An option provides the holder with the right to
  buy or sell a specified quantity of an underlying
  asset at a fixed price (called a strike price or
  an exercise price) at or before the expiration
  date of the option.
• There has to be a clearly defined underlying
  asset whose value changes over time in
  unpredictable ways.
• The payoffs on this asset (real option) have to
  be contingent on an specified event occurring
  within a finite period.

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A Call Option

• A call option gives you the right to buy an
  underlying asset at a fixed price (called a
  strike or an exercise price).
• That right may extend over the life of the option
  (American option) or may apply only at the end
  of the period (European option).

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Payoff Diagram on a Call
Net Payoff
on Call
Strike
Price
Price of underlying asset
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A Put Option

• A put option gives you the right to buy an
  underlying asset at a fixed price (called a
  strike or an exercise price).
• That right may extend over the life of the option
  (American option) or may apply only at the end
  of the period (European option).

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Payoff Diagram on Put Option
Net Payoff On Put
Strike Price
Price of underlying asset
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Determinants of option value

• Variables Relating to Underlying Asset
• Value of Underlying Asset as this value
  increases, the right to buy at a fixed price
  (calls) will become more valuable and the right
  to sell at a fixed price (puts) will become less
  valuable.
• Variance in that value as the variance
  increases, both calls and puts will become more
  valuable because all options have limited
  downside and depend upon price volatility for
  upside.
• Expected dividends on the asset, which are likely
  to reduce the price appreciation component of the
  asset, reducing the value of calls and increasing
  the value of puts.
• Variables Relating to Option
• Strike Price of Options the right to buy (sell)
  at a fixed price becomes more (less) valuable at
  a lower price.
• Life of the Option both calls and puts benefit
  from a longer life.
• Level of Interest Rates as rates increase, the
  right to buy (sell) at a fixed price in the
  future becomes more (less) valuable.

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The essence of option pricing models The
Replicating portfolio Arbitrage

• Replicating portfolio Option pricing models are
  built on the presumption that you can create a
  combination of the underlying assets and a risk
  free investment (lending or borrowing) that has
  exactly the same cash flows as the option being
  valued. For this to occur,
• The underlying asset is traded - this yield not
  only observable prices and volatility as inputs
  to option pricing models but allows for the
  possibility of creating replicating portfolios
• An active marketplace exists for the option
  itself.
• You can borrow and lend money at the risk free
  rate.
• Arbitrage If the replicating portfolio has the
  same cash flows as the option, they have to be
  valued (priced) the same.

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Creating a replicating portfolio

• The objective in creating a replicating portfolio
  is to use a combination of riskfree
  borrowing/lending and the underlying asset to
  create the same cashflows as the option being
  valued.
• Call Borrowing Buying D of the Underlying
  Stock
• Put Selling Short D on Underlying Asset
  Lending
• The number of shares bought or sold is called the
  option delta.
• The principles of arbitrage then apply, and the
  value of the option has to be equal to the value
  of the replicating portfolio.

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The Binomial Option Pricing Model
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The Limiting Distributions.

• As the time interval is shortened, the limiting
  distribution, as t -gt 0, can take one of two
  forms.
• If as t -gt 0, price changes become smaller, the
  limiting distribution is the normal distribution
  and the price process is a continuous one.
• If as t-gt0, price changes remain large, the
  limiting distribution is the poisson
  distribution, i.e., a distribution that allows
  for price jumps.
• The Black-Scholes model applies when the limiting
  distribution is the normal distribution , and
  explicitly assumes that the price process is
  continuous and that there are no jumps in asset
  prices.

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Black and Scholes to the rescue

• The version of the model presented by Black and
  Scholes was designed to value European options,
  which were dividend-protected.
• The value of a call option in the Black-Scholes
  model can be written as a function of the
  following variables
• S Current value of the underlying asset
• K Strike price of the option
• t Life to expiration of the option
• r Riskless interest rate corresponding to the
  life of the option
• ?2 Variance in the ln(value) of the underlying
  asset

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The Black Scholes Model

• Value of call S N (d1) - K e-rt N(d2)
• where
• d2 d1 - ? vt
• The replicating portfolio is embedded in the
  Black-Scholes model. To replicate this call, you
  would need to
• Buy N(d1) shares of stock N(d1) is called the
  option delta
• Borrow K e-rt N(d2)

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The Normal Distribution
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Adjusting for Dividends

• If the dividend yield (y dividends/ Current
  value of the asset) of the underlying asset is
  expected to remain unchanged during the life of
  the option, the Black-Scholes model can be
  modified to take dividends into account.
• Call value S e-yt N(d1) - K e-rt N(d2)
• where,
• d2 d1 - ? vt
• The value of a put can also be derived from
  put-call parity (an arbitrage condition)
• Put value K e-rt (1-N(d2)) - S e-yt (1-N(d1))

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Choice of Option Pricing Models

• Some practitioners who use option pricing models
  to value options argue for the binomial model
  over the Black-Scholes and justify this choice by
  noting that
• Early exercise is the rule rather than the
  exception with real options
• Underlying asset values are generally
  discontinous.
• In practice, deriving the end nodes in a binomial
  tree is difficult to do. You can use the variance
  of an asset to create a synthetic binomial tree
  but the value that you then get will be very
  similar to the Black Scholes model value.